Two New Generalized Radial Epiderivatives in Set-Valued Analysis

Abstract

The generalized radial epiderivative for nonconvex set-valued maps is formulated in terms of minimal points and therefore does not separately reflect weak and Benson proper minimality. To extend this framework to these two efficiency concepts, we introduce the weak generalized radial epiderivative and the proper generalized radial epiderivative by using weakly minimal and Benson properly minimal points, respectively. We establish existence results for generalized radial epiderivatives under regularity and C-lower-boundedness assumptions and investigate the relationships among the generalized, weak generalized, proper generalized, and radial epiderivatives. In particular, conditions under which the generalized radial epiderivative coincides with the radial epiderivative are obtained. We also provide scalarization-based characterizations of the weak and proper generalized radial epiderivatives by means of augmented dual cones and strongly monotonically increasing functionals. The examples illustrate that the proposed weak and proper notions may differ from the generalized radial epiderivative, showing that they provide distinct extensions of the existing framework.

1. Introduction

Set-valued analysis provides a fundamental framework for the study of generalized differentiation and optimization of set-valued maps [1]. The basic theory of vector optimization and ordered linear spaces has also been developed extensively in [2,3]. Early optimality conditions in partially ordered linear spaces and set-valued optimization were investigated in [4]. Within this framework, Jahn and Rauh introduced the contingent epiderivative for set-valued maps [5], while Chen and Jahn generalized this concept [6]. Jahn and Khan subsequently established existence theorems and characterizations for generalized contingent epiderivatives [7], and further related developments concerning contingent derivatives of set-valued maps were presented by Luc [8].

Radial approaches were subsequently developed for nonconvex optimization. Optimality conditions in nonconvex set-valued optimization were studied by Flores-Bazán [9], who later investigated radial epiderivatives and asymptotic functions in nonconvex vector optimization [10]. Kasimbeyli studied radial epiderivatives in set-valued optimization [11], while Kasimbeyli and Mammadov investigated the relationships among weak subdifferentials, directional derivatives, and radial epiderivatives for nonconvex functions [12]. The generalized radial epiderivative for set-valued maps was introduced by Kasimbeyli and İnceoğlu [13], where existence conditions and optimality conditions were established. More recently, Yalcin and Kasimbeyli investigated generalized derivatives and optimality conditions in nonconvex optimization [14].

The generalized radial epiderivative introduced in [13] is defined in terms of minimal points of the corresponding radial set. However, in set-valued and vector optimization, weak minimality and Benson proper minimality represent distinct efficiency concepts and need not coincide with ordinary minimality. Consequently, derivative constructions based on these notions can capture weakly efficient and properly efficient behavior that is not reflected by the generalized radial epiderivative based only on minimal points. Motivated by this distinction, we introduce the weak generalized radial epiderivative and the proper generalized radial epiderivative by employing weakly minimal points and Benson properly minimal points, respectively. Although these notions are constructed within the generalized radial epiderivative framework, they lead, in general, to different derivative sets and extend the radial approach to weak and proper efficiency concepts.

The main contributions of this paper are the introduction of these two generalized radial epiderivatives, the establishment of existence and inclusion results, and the derivation of scalarization-based characterizations using augmented dual cones and strongly monotonically increasing functionals. The examples further illustrate that the proposed weak and proper generalized radial epiderivatives need not coincide with the existing generalized radial epiderivative. Thus, the present work extends the framework of [13] to weak and proper minimality rather than merely reformulating the existing concept.

2. Preliminaries

We recall the basic concepts together with some standard notions.

Definition 1 

([11,12]). Let A be a nonempty subset of a real normed space (X,‖.‖_X). The closed radial cone $R\left(A,\overline{x}\right)$ of A at $\overline{x}\in \mathit{cl}\left(A\right)$ is the set of all x ∈ X such that there exist δ_n > 0 and a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}\subset X$ with $\underset{n\to \infty }{\mathit{lim}}{x}_n=x$ so that $\overline{x}+{\delta }_n{x}_n\in A$ for all $n\in \mathbb{N}$.

Note that the radial cone can equivalently be characterized as follows.

Definition 2 

([11,12]). Let A be a nonempty subset of a real normed space (X,‖.‖_X). The closed radial cone $R\left(A,\overline{x}\right)$of A at $\overline{x}\in \mathit{cl}\left(A\right)$is the set of all x ∈ X such that there exist δ_n > 0 and a sequence ${\left(x_n\right)}_{n\in \mathbb{N}}\subset A$ with $\underset{n\to \infty }{\mathit{lim}}{\delta }_n\left(x_n-\overline{x}\right)=x.$

It follows from this definition that

$$R\left(A,\overline{x}\right)=\mathit{cl}\left(\mathit{cone}\left(A-\overline{x}\right)\right)$$

(1)

where cone denotes the conic hull of a set, which is the smallest cone containing $A-\overline{x}.$

Definition 3 

([11,12,13]). Let (X,‖.‖_X) and (Y,‖.‖_Y) be real normed spaces, let F:X → 2^Y be a set-valued map.

(a) 

The set

$$\mathit{graph}\left(F\right)=\left\{\left(x,y\right)\in X\times Y|x\in X,y\in F\left(x\right)\right\}$$

(2)

is called the graph of F

(b) 

The set

$$\mathit{dom}\left(F\right)=\left\{x\in X|F\left(x\right)\ne \varnothing \right\}\hspace{1em}$$

(3)

is called the domain of F. F is said to be proper if dom(F) ≠ ∅

(c) 

Let Y be partially ordered by a convex cone C ⊂ Y. The set

$$\mathit{epi}\left(F\right)=\left\{\left(x,y\right)\in X\times Y|x\in X,y\in F\left(x\right)+C\right\}$$

(4)

is called the epigraph of F.

(d) 

Let F: X → 2^Y be a set-valued map. The set-valued map P_F:X → 2^Y defined, for every x ∈ dom(F)

$$P_F\left(x\right)=F\left(x\right)+C$$

(5)

is called the profile map of F at x ∈ dom(F).

Definition 4 

([11,12]). Let (X,‖.‖_X) and (Y,‖.‖_Y) be real normed spaces, let Y be partially ordered by a convex cone C ⊂ Y, let S be nonempty subset of X and let F:S → 2^Y be a set-valued map. Let a pair $\left(\overline{x},\overline{y}\right)\in \mathit{graph}\left(F\right)$  be given. Set H:= {u ∈ X: there exists v ∈ Y such that (u, v) ∈ R(epi(F), ($\overline{x}$, $\overline{y}$))}. A single-valued map $D_{r}F\left(\overline{x},\overline{y}\right):H\to Y$ whose epigraph equals the radial cone to the epigraph of F at $\left(\overline{x},\overline{y}\right)$, i.e.,

$$\mathit{epi}\left(D_{r}F\left(\overline{x},\overline{y}\right)\right)=R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right)$$

(6)

is called the radial epiderivative of $F \mathit{at} \left(\overline{x},\overline{y}\right).$

Definition 5 

([11]). Let (X,‖.‖_X) and (Y,‖.‖_Y) be real normed spaces, let Y be partially ordered by a convex cone C ⊂ Y, let S be nonempty subset of X and let F:S → 2^Y be a set-valued map. Let a pair $\left(\overline{x},\overline{y}\right)\in \mathit{graph}\left(F\right)$ be given. A set-valued map $D_{R}F\left(\overline{x},\overline{y}\right):X\to 2^Y$ whose graph equals the radial cone to the graph of F at $\left(\overline{x},\overline{y}\right)$, i.e.,

$$\mathit{graph}\left(D_{R}F\left(\overline{x},\overline{y}\right)\right)=R\left(\mathit{graph}\left(F\right),\left(\overline{x},\overline{y}\right)\right)$$

(7)

is called the radial derivative of $F \mathit{at} \left(\overline{x},\overline{y}\right).$

Definition 6. 

Let (Y,‖.‖_Y) be a real normed space partially ordered by a pointed convex cone C ⊂ Y. Let D be a nonempty subset of Y and let d ∈ D be arbitrary.

(a) 

The element d is said to be a minimal point of D, if D ∩ ({d} − C) = {d}.

(b) 

The element d is said to be a strongly minimal point of D, if D ⊆ {d} + C

(c) 

Let the ordering cone have a nonempty interior int(C). The element d is said to be a weakly minimal point of D, if D ∩ ({d} − int(C)) = ∅.

(d) 

The element d is said to be a properly minimal point of D (in the sense of Benson), if d is a minimal element of D and the zero point of Y is a minimal point of the set R(D + C,d) = cl(cone(D + C − {d})), that is, R (D + C,d) ∩ (−C) = {0_Y} [11,15].

The set of all minimal points, strongly minimal points, weakly minimal points, properly minimal points of the set D with respect to a cone C will be denoted by min(D,C),s − min(D,C), w − min(D,C), pr − min(D,C), respectively. If D = ∅, we define min(D,C) = ∅.

Definition 7 

([16]). Let (Y,‖.‖_Y) be a real normed space partially ordered by a pointed convex cone C ⊂ Y

(a) 

The cone C is Daniell, if any decreasing sequence in Y having a lower bound converges to its infimum.

(b) 

A subset B is said to be minorized, if there is a y ∈ Y so that B ⊂ {y} + C.

(c) 

The domination property is said to hold for a subset B of Y if B ⊂ min(B,C) + C

(d) 

A convex cone C is called regular if every C − decreasing and C − lower bounded sequence converges to an element of Y.

3. Results

The following concept, introduced in [13], is a generalization of the radial epiderivative concept given for a special case in [11].

Definition 8. 

A set-valued map $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right): X\to 2^Y$ is called the generalized radial epiderivative of F at $\left(\overline{x},\overline{y}\right),$ if

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)=\mathit{min}\left(G\left(x\right),C\right),$$

(8)

where G: X → 2^Y is the set-valued map given by

$$G\left(x\right)=\left\{y\in Y:\left(x,y\right)\in R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right)\right\},\forall x\in X.$$

(9)

It is easy to see that G(x) is a closed set and G(x) = G(x) + C.

If we write w − min and pr − min instead of min, we define the weak generalized radial epiderivative and the proper generalized radial epiderivative, respectively. We denote these epiderivatives by $D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)$ and $D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)$.

Because of the inclusions

$$\mathit{pr}-\mathit{min}\left(.,.\right)\subseteq \mathit{min}\left(.,.\right)\subseteq w-\mathit{min}\left(.,.\right)$$

(10)

the following inclusions are obvious

$$D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(x\right)\subseteq D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)\subseteq D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$$

(11)

Remark 1. 

In the equality (8) min denotes the set of minimal points of a set G(x) with respect to cone C.

The following example illustrates that the generalized radial epiderivative and the proper generalized radial epiderivative may differ, even for a simple nonconvex set-valued map.

Example 1. 

Let $F:\mathbb{R}\to 2^{{\mathbb{R}}^2}$ be a set-valued map defined by

$$F\left(x\right)=\left\{\begin{array}{r}\varnothing ,\hspace{1em}\mathit{if} x<0\\ (y_1,y_2)\in {\mathbb{R}}^2:{y_1}^2+{y_2}^2\le x^2,\hspace{1em}\mathit{if} x\ge 0\end{array}\right.$$

Let $\left(\overline{x},\overline{y}\right)=\left(0,(0,0)\right)$, let B denote the closed unit ball of ${\mathbb{R}}^2$. For x ≥ 0, we have

$$F\left(x\right)=\mathit{xB}.$$

Since

$$\mathit{epi}\left(F\right)=\left\{\left(x,y\right)\in \mathbb{R}\times {\mathbb{R}}^2:x\ge 0,y\in \mathit{xB}+{\mathbb{R}}_{+}^2\right\},$$

and this set is a closed cone with vertex (0,(0,0)), its radial cone at $\left(\overline{x},\overline{y}\right)$ coincides with itself. Therefore

$$G\left(x\right)=\mathit{xB}+{\mathbb{R}}_{+}^2=F\left(x\right)+{\mathbb{R}}_{+}^2,\hspace{1em}x\ge 0.$$

The minimal points of G(x) with respect to $C={\mathbb{R}}_{+}^2$are the points on the lower-left boundary of the disk xB. Hence,

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)≔\left\{(y_1,y_2)\in {\mathbb{R}}^2:{y_1}^2+{y_2}^2=x^2,y_1\le 0,y_2\le 0\right\},\hspace{1em}x>0.$$

For $\overline{x}=0$ and  $\overline{y}=(0,0$) we have

$$R\left(\mathit{epi}\right(F),(0,(0,0)\left)\right)=\left\{(u,v)\in \mathbb{R}\times {\mathbb{R}}^2:u\ge 0,v\in \mathit{uB}+{\mathbb{R}}_{+}^2\right\}.$$

Moreover, by the definition of the generalized radial epiderivative

$$D_{\mathit{gr}}F\left(0,(0,0)\right)\left(u\right)=\left\{y\in {\mathbb{R}}^2:‖y‖=u\ge 0,y_1\le 0,y_2\le 0\right\},\hspace{1em}u\ge 0,$$

and therefore

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(0,(0,0)\right)\right)=\left\{(u,v)\in {\mathbb{R}\times \mathbb{R}}^2:u\ge 0,v\in \mathit{uB}+{\mathbb{R}}_{+}^2\right\}.$$

Since $\mathit{uB}+{\mathbb{R}}_{+}^2=\left\{y\in {\mathbb{R}}^2:‖y‖=u\ge 0,y_1\le 0,y_2\le 0\right\}+{\mathbb{R}}_{+}^2$, we have

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(0,(0,0)\right)\right)=R\left(\mathit{epi}\left(F\right),\left(0,(0,0)\right)\right)$$

Now, we determine the proper generalized radial epiderivative. Let x = 1. Then

$$D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(1\right)≔\left\{(y_1,y_2)\in {\mathbb{R}}^2:{y_1}^2+{y_2}^2=1,y_1<0,y_2<0\right\}$$

Consider the point d = (−1,0). Although d is a minimal point of G(1), it is not a properly minimal point in the sense of Benson. Indeed, by Definition 6(d), proper minimality requires

$$R\left(G\left(1\right)+C,d\right)\cap (-C)=\left\{0\right\}.$$

Since

$$G\left(1\right)=F\left(1\right)+C,$$

we have

$$G\left(1\right)+C=F\left(1\right)+C+C=F\left(1\right)+C=G\left(1\right),$$

and therefore

$$R\left(G\left(1\right)+C,d\right)=R\left(G\left(1\right),\mathrm{d}\right).$$

For d = (−1,0), this radial cone contains a nonzero direction belonging to −C. Consequently

$$R\left(G\left(1\right)+C,d\right)\cap (-C)\ne \left\{0\right\}$$

and hence

$$\left(-1,0\right)\notin D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(1\right)$$

Similarly,

$$\left(0,-1\right)\notin D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(1\right)$$

We verify that these points are properly minimal in the sense of Benson. Let d = (d₁,d₂) be such that d₁ < 0, d₂ < 0 and d₁²+d₂²=1.

Set p = −d ∈ intC. For every $z\in G\left(1\right)=F\left(1\right)+{\mathbb{R}}_{+}^2$, there exist q ∈ F(1) and c ∈ C such that z = q + c. Hence, by the Cauchy–Schwarz inequality

$$⟨p,z⟩=⟨p,q⟩+⟨p,c⟩\ge -‖p‖‖q‖\ge -1=⟨p,d⟩.$$

Now, let

$$w\in R\left(G\right(1)+C,d)\cap (-C).$$

Since G(1) + C = G(1), we also have

$$w\in R\left(G\right(1),d)\cap (-C).$$

By the definition of the radial cone, there exist λ_n > 0 and w_n → w such that

$$d+{\lambda }_n{w}_n\in G\left(1\right)$$

for every $n\in \mathbb{N}$. Therefore,

$$⟨p,d+{\lambda }_n{w}_n⟩\ge ⟨p,d⟩,$$

which implies

$$⟨p,w_n⟩\ge 0.$$

Taking n → ∞, we get

$$⟨p,w⟩\ge 0.$$

Since p ∈ intC and w ∈ −C we also have

$$⟨p,w⟩\le 0.$$

Thus ⟨p,w⟩ = 0 and, hence w = (0,0). Consequently

$$R\left(G\right(1),d)\cap (-{\mathbb{R}}_{+}^2)=\left\{\left(0,0\right)\right\}.$$

This proves that d is properly minimal in the sense of Benson.

Consequently,

$$D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(1\right)=\left\{(y_1,y_2)\in {\mathbb{R}}^2:{y_1}^2+{y_2}^2=1,y_1<0,y_2<0\right\}$$

whereas

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(1\right)=\left\{(y_1,y_2)\in {\mathbb{R}}^2:{y_1}^2+{y_2}^2=1,y_1\le 0,y_2\le 0\right\}$$

Thus

$$D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(1\right)⊊D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(1\right).$$

This example shows explicitly that the proper generalized radial epiderivative may be strictly smaller than the generalized radial epiderivative.

The following example shows that the weak generalized radial epiderivative may be strictly larger than the generalized radial epiderivative.

Example 2. 

Let C = ℝ₊². Let $F:\mathbb{R}\to 2^{{\mathbb{R}}^2}$ be a set-valued map defined by

$$F\left(x\right)=\left\{\begin{array}{r}\varnothing , \mathit{if} x<0\\ \left\{\left(0,0\right)\right\}, \mathit{if} x\ge 0\end{array}\right.$$

Consider $\left(\overline{x},\overline{y}\right)=\left(0,(0,0)\right)$. Then

$$\mathit{epiF}=\left\{\left(x,y\right)\in \mathbb{R}\times {\mathbb{R}}^2:x\ge 0,y\in C\right\}.$$

Since this set is a closed cone with vertex (0,(0,0)), its radial cone at $\left(\overline{x},\overline{y}\right)$ coincides with itself. Hence, for every x ≥ 0,

$$G\left(x\right)=C.$$

Therefore,

$$\mathit{min}\left(G\left(x\right),C\right)=\left\{\left(0,0\right)\right\},$$

and consequently

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)=\left\{\left(0,0\right)\right\}.$$

On the other hand, since $\mathit{int}\left(C\right)=\left\{\left(y_1,y_2\right)\in {\mathbb{R}}^2:y_1>0,y_2>0\right\}$,

$$w-\mathit{min}\left(G\left(x\right),C\right)=\left({\mathbb{R}}_{+}\times \left\{0\right\}\right)\cup \left(\left\{0\right\}\times {\mathbb{R}}_{+}\right).$$

Thus,

$$D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)=\left({\mathbb{R}}_{+}\times \left\{0\right\}\right)\cup \left(\left\{0\right\}\times {\mathbb{R}}_{+}\right),$$

whereas

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)=\left\{\left(0,0\right)\right\}.$$

Hence,

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)⊊D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right).$$

The following lemma collects two results concerning the existence of minimal points and the domination property; see [17] (Proposition 3.4 and Theorem 5.14).

Lemma 1. 

Let (Y,‖.‖_Y) be a real normed space and let D ⊂ Y be nonempty. Then,

(a) 

If the cone C is regular and D is closed and C-lower bounded, then min(D,C) ≠ ∅ and the domination property holds, that is, D ⊂ min(D,C) + C.

(b) 

Let the cone C have a compact base and let D be closed and convex. If min(D,C) ≠ ∅, then the domination property holds.

Theorem 1. 

Let (X,‖.‖_X) and (Y,‖.‖_Y) be real normed spaces, let C ⊂ Y be a regular pointed convex cone, let F:X → 2^Y be a set-valued map. Let a pair $\left(\overline{x},\overline{y}\right)\in \mathit{graph}\left(F\right)$ be given. Suppose that, for every x ∈ dom(G), the set G(x) has a C − lower bound. Then for every x ∈ dom(G), the generalized radial epiderivative $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$ exists. Moreover, the following relation holds:

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)=R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right)$$

(12)

If int(C) ≠ ∅, then the weak generalized radial epiderivative $D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$ also exists and satisfies

$$\mathit{epi}\left(D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\right)=R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right)$$

(13)

Proof. 

Since the radial cone is closed in a real normed space, the set G(x) is closed for each x ∈ dom(G). Moreover, G(x) has a lower bound by the assumption. Hence, since C is regular, all the hypotheses of Lemma 1(a) are satisfied. Therefore, min(G(x),C) ≠ ∅, $\mathit{so} { D}_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)$ exists and the domination property holds. Then, we have

$$G\left(x\right)\subseteq D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)+C.$$

(14)

To prove the reverse inclusion, let $v\in D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)+C$. Since

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)=\mathit{min}\left(G\right(x),C)\subseteq G\left(x\right)$$

and

$$G\left(x\right)+C=G\left(x\right),$$

it follows that

$$v\in G\left(x\right).$$

Hence,

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)+C\subseteq G\left(x\right).$$

Combining this inclusion with (14), we obtain that

$$G\left(x\right)=D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)+C.$$

By the definitions of the radial derivative and the profile map P_F of F, we get

$$\begin{array}{cc}{D}_R{P}_F\left(\overline{x},\overline{y}\right)\left(x\right)=& \left\{y\in Y:\left(x,y\right)\in R\left(\mathit{graph}\left(P_F\right),\left(\overline{x},\overline{y}\right)\right)\right\}\\ =& \left\{y\in Y:\left(x,y\right)\in R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right)\right\}\\ =& G\left(x\right)\end{array}$$

and then

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)+C=G\left(x\right)=D_R{P}_F\left(\overline{x},\overline{y}\right)\left(x\right).$$

(15)

By (15) and the definition of the epigraph, we obtain

$$\begin{array}{c}\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)\\ \\ \end{array}\begin{array}{c}=\\ =\\ =\end{array}\begin{array}{c}\mathit{graph}\left(D_R{P}_F\left(\overline{x},\overline{y}\right)\right)\\ R\left(\mathit{graph}\left(P_F\right),\left(\overline{x},\overline{y}\right)\right)\\ R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right)\end{array}$$

Thus (12) is proven.

Now let us prove (13). Since $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$ exists and $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)\subseteq D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right),$ $D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$ exists. It follows from (9) and (14) that $G\left(x\right)\subseteq D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)+C$. The reverse inclusion is similar to the proof of (12). The proof is completed. □

The reason why identities (12) and (13) play a fundamental role in this approach is that various properties of radial cones are used to obtain various conclusions about the corresponding epiderivatives. The domination property is an important characteristic for understanding the relationship between epiderivatives.

If the weak generalized radial epiderivative exists, the proof is made by assuming the weak domination property G(x) ⊆ w − min(G(x),C) + C.

Corollary 1. 

Let C ⊂ Y be a regular pointed convex cone with int(C) ≠ ∅. Let F:X → 2^Y be a set-valued map and let $(\overline{x},\overline{y})\in \mathit{graph}(F$). Suppose that, for every x ∈ dom(G), the set G(x), given by (9), has a C-lower bound. Then

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)=\mathit{epi}\left(D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\right).$$

(16)

Proof. 

According to (12) given in Theorem 1, we know that

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)=R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right).$$

On the other hand, with the same theorem assumptions

$$\mathit{epi}\left(D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\right)=R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right).$$

Thus, we have

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)=\mathit{epi}\left(D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\right).$$

□

Corollary 2. 

Let F:X → 2^Y be a set-valued map and $\left(\overline{x},\overline{y}\right)\in \mathit{graph}\left(F\right)$, let C ⊂ Y be a regular pointed convex cone with intC ≠ ∅. Suppose that, for every x ∈ dom(G), the set G(x), given by (9), has a C-lower bound. The following holds:

$$\mathit{epi}\left(D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\right)\subseteq R\left(\mathit{dom}\left(F\right),\overline{x}\right)\times Y$$

(17)

and

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)\subseteq R\left(\mathit{dom}\left(F\right),\overline{x}\right)\times Y$$

(18)

Proof. 

Let $\left(u,v\right)\in \mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)$ . By (12), $\left(u,v\right)\in R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right).$ By the definition of the radial cone, there exist sequences λ_n > 0, (u_n,v_n) → (u,v) such that

$$\left(\overline{x}+{\lambda }_n{u}_n,\overline{y}+{\lambda }_n{v}_n\right)\in \mathit{epiF}$$

for every $n\in \mathbb{N}$. Therefore,

$$\overline{y}+{\lambda }_n{v}_n\in F\left(\overline{x}+{\lambda }_n{u}_n\right)+C,$$

in particular

$$\overline{x}+{\lambda }_n{u}_n\in \mathit{dom}\left(F\right).$$

Since u_n → u, the sequence ${\left(u_n\right)}_{n\in N}$ together with λ_n > 0 belongs to the radial cone of dom(F) at $\overline{x}$. Thus,

$$u\in R\left(\mathit{dom}\right(F),\overline{x}).$$

Consequently, we obtain that

$$(u,v)\in R\left(\mathit{dom}\left(F\right),\overline{x}\right)\times Y.$$

Hence,

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)\subseteq R\left(\mathit{dom}\left(F\right),\overline{x}\right)\times Y$$

By (16),

$$\mathit{epi}\left(D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\right)\subseteq R\left(\mathit{dom}\left(F\right),\overline{x}\right)\times Y.$$

This completes the proof. □

The following corollary shows that if the ordering cone C is pointed and the radial epiderivative exists then this coincides with the generalized radial epiderivative.

Corollary 3. 

Let X and Y be real normed spaces, let C ⊂ Y be a regular pointed convex cone, let F:X → 2^Y be a set-valued map. Let a pair$\left(\overline{x},\overline{y}\right)\in \mathit{graph}\left(F\right)$ be given. Suppose that, for every x ∈ dom(G), the set G(x) has a C − lower bound and s − min(G(x),C) ≠ ∅. Then the radial epiderivative $D_{r}F\left(\overline{x},\overline{y}\right)$ and the generalized radial epiderivative $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)$ coincide on dom(G), identifying each singleton value with its unique element.

Proof. 

Let x ∈ dom(G) and let d ∈ s − min(G(x),C). By the definition of a strongly minimal point,

$$G\left(x\right)\subseteq \left\{d\right\}+C.$$

We first show that d ∈ min(G(x),C). Let y ∈ G(x) satisfy

$$d-y\in C.$$

Since y ∈ G(x) ⊆ {d} + C, we also have y − d ∈ C.

Since the cone C is pointed, C ∩ (−C) = {0_Y}, and therefore y = d. Hence

$$d\in \mathit{min}\left(G\left(x\right),C\right).$$

Now, let m ∈ min(G(x),C). Since m ∈ G(x) ⊆ {d} + C, we have m − d ∈ C.

Since d ∈ G(x) and m is a minimal point of G(x), it follows that m = d. Consequently,

$$\mathit{min}\left(G\left(x\right),C\right)=\left\{d\right\}.$$

Hence, $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)$ is a single-valued map on dom(G). Moreover, by Theorem 1

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)=R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right).$$

Therefore, by Definition 4, $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)$ is the radial epiderivative of F at $\left(\overline{x},\overline{y}\right)$.

Hence

$$D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)=D_{r}F\left(\overline{x},\overline{y}\right)$$

The uniqueness of the radial epiderivative also follows from Theorem 3.5 in [11]. This completes the proof. Outside dom(G), the generalized radial epiderivative has empty values. □

Definition 9 

([18]). A set-valued map F:X → 2^Y is called C− lower semicontinuous, if the set L_y(x)≔{x ∈ X:F(x) ∩ (y − C) ≠ ∅} is closed for all y ∈ Y.

The following corollary shows that the generalized radial epiderivative is C− lower semicontinuous.

Corollary 4. 

Let C ⊂ Y be a regular pointed convex cone with intC ≠ ∅. Let F:X → 2^Y be a set-valued map and let $\left(\overline{x},\overline{y}\right)\in \mathit{graph}\left(F\right)$. Suppose that dom(G) = X and, for every x ∈ X, the set G(x) has a C − lower bound. Then $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)$ and $D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)$ are C− lower semicontinuous.

Proof. 

We prove only that $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)$ is a C− lower semicontinuous map since the C− lower semicontinuity of $D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)$ follows by a similar argument. Let y ∈ Y. By Theorem 1

$$\mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)=R\left(\mathit{epi}\left(F\right),\left(\overline{x},\overline{y}\right)\right),$$

and the latter set is closed. Moreover,

$$\left\{x\in X:D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)\cap \left(y-C\right)\ne \varnothing \right\}=\left\{x\in X:(x,y)\in \mathit{epi}\left(D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\right)\right\},$$

it follows from the closedness of the radial cone that for every y ∈ Y the set

$$\{x\in X:D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)\left(x\right)\cap (y-C)\ne \varnothing \}$$

is closed. Therefore, the generalized radial epiderivative $D_{\mathit{gr}}F\left(\overline{x},\overline{y}\right)$ is a C− lower semicontinuous map. This completes the proof. □

Characterization

Let (Y,‖.‖_Y) be a real normed space partially ordered by a closed, pointed, convex cone C⊂Y.

The dual cone and its quasi-interior are defined by:

$$\begin{gathered} C^{\star }=\left\{l\in Y^{\star }:l\left(x\right)\ge 0; \forall x\in C\right\}, \\ {C}^{⧣}=\left\{l\in Y^{\star }:l\left(x\right)>0, \forall x\in C╲0_Y\right\} \end{gathered}$$

The generalized forms of these definitions, given in [19,20] and called augmented dual cones, are defined as follows:

$$\begin{gathered} U=\left\{(l,\alpha )\in C^{⧣}\times {\mathbb{R}}_{+}:\alpha ‖y‖-l\left(y\right)\le 0,\forall y\in C\right\} \\ {U}^{⧣}=\left\{(l,\alpha )\in C^{⧣}\times {\mathbb{R}}_{+}:\alpha \Vert y\Vert -l\left(y\right)<0,\forall y\in C╲0_Y\right\} \end{gathered}$$

Let (l,α) ∈ U^⧣. Define the following sets:

$$\begin{gathered} G_{\left(l,\alpha \right)}\left(x\right)=\left\{\alpha ‖y‖+l\left(y\right)\in \mathbb{R}:y\in G\left(x\right)\right\} \\ S{G}_{\left(l,\alpha \right)}\left(x\right)=\left\{z\in G\left(x\right): \alpha ‖z‖+l\left(z\right)\le \alpha ‖y‖+l\left(y\right), \forall y\in G\left(x\right)\right\} \end{gathered}$$

In other words, SG_(l,α)(x) consists of all $\overline{y}\in Y$ belonging to G(x) such that the scalarized value $\alpha ‖\overline{y}‖+l\left(\overline{y}\right)$ is not greater than the scalarized value of any element of G(x).

We use the following notations defined in [19]:

Let C be a closed convex pointed cone (i.e., C ∩ (−C) = {0_Y}) with nonempty interior int(C), for any y, d ∈ Y,

$$\begin{array}{ccc}y{\leqq }_{C}d & \mathit{iff}& d-y\in C\\ y{\le }_{C}d& \mathit{iff}& d-y\in C\setminus \left\{0\right\}\\ y{<}_{C}d& \mathit{iff}& d-y\in \mathit{int}\left(C\right)\end{array}$$

Definition 10 

([19]). Let Y be an ordered real normed space with a closed pointed convex cone C having nonempty interior int(C), and let D be a nonempty subset of Y, and let $g:D\to \mathbb{R}$ be a functional. For each d ∈ D

(a) 

if y ∈ ({d} − C) ∩ D ⇒ g(y) ≤ g(d), then the functional g is called monotonically increasing,

(b) 

if y ∈ ({d} − C) ∩ D, y ≠ d ⇒ g(y) < g(d), then the functional g is called strongly monotonically increasing.

Lemma 2 

([19]). Let $g:Y\to \mathbb{R}$ be a function defined by

$$g\left(y\right)=\alpha \Vert y\Vert +l\left(y\right)$$

where (l,α) ∈ U ((l,α) ∈ U^⧣). Then g is monotonically increasing (strongly monotonically increasing).

Theorem 2. 

Let (X,‖.‖_X) and (Y,‖.‖_Y) be real normed spaces, let C ⊂ Y be a closed pointed convex cone, let F:X → 2^Y be a set-valued map. Let a pair $\left(\overline{x},\overline{y}\right)\in \mathit{graph}\left(F\right)$ be given. Assume that for some x ∈ X the proper generalized radial epiderivative $D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$ exists. Then

$$\bigcup _{(l,\alpha )\in U^{⧣}}{\mathit{SG}}_{(l,\alpha )}\left(x\right)\subseteq D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$$

Proof. 

Let $y_0\in \bigcup _{\left(l,\alpha \right)\in U^{⧣}}S{G}_{\left(l,\alpha \right)}\left(x\right)$ . Then there exists (l,α) ∈ U^⧣ such that for every y ∈ G(x)

$$\alpha ‖y_0‖+l\left(y_0\right)\le \alpha ‖y‖+l\left(y\right)$$

From here for every y ∈ G(x), we obtain

$$\alpha ‖y‖+l\left(y\right)-\alpha ‖y_0‖-l\left(y_0\right)\ge 0$$

and therefore, by the triangle inequality, for every y ∈ G(x)

$$\alpha ‖y-y_0‖+l(y-y_0)\ge 0$$

(19)

For every z ∈ cone(G(x) + C − {y₀}), there exist y ∈ G(x), c ∈ C and λ ≥ 0 such that

$$z=\lambda \left(y+c-y_0\right).$$

Then, by Lemma 2, we have

$$\begin{gathered} \alpha ‖z‖+l\left(z\right)=\lambda (\alpha ‖y-y_0+c‖+l\left(y-y_0+c\right)) \\ \ge \lambda (\alpha ‖y-y_0‖+l\left(y-y_0\right))\ge 0. \end{gathered}$$

Thus, for every z ∈ cone(G(x) + C − {y₀}), α‖z‖ + l(z) ≥ 0.

For an arbitrary fixed y ∈ cl(cone(G(x) + C − {y₀})), there exists a sequence

$${\left(y_n\right)}_{n\in \mathbb{N}}\subset \mathit{cone}\left(G\left(x\right)+C-\left\{y_0\right\}\right),$$

with $y=\underset{n\to \infty }{\lim }{y}_n$. By the preceding inequality, we have for all $n\in \mathbb{N},$

$$\alpha ‖y_n‖+l\left(y_n\right)\ge 0.$$

Taking n → ∞, we obtain for all y ∈ cl(cone(G(x) + C − {y₀})):

$$\alpha ‖y‖+l\left(y\right)\ge 0$$

(20)

Since g(y) = α‖y‖ + l(y) is strongly monotonically increasing and y₀ minimizes g over G(x), it follows that

$$y_0\in \mathit{min}\left(G\left(x\right),C\right).$$

Now, suppose that $y_0\notin D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$. Since $y_0\notin D_{\mathit{pr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$, by Definition 6(d), there exists y^* ∈ Y such that

$$y^{*}\in \mathit{cl}\left(\mathit{cone}\left(G\left(x\right)+C-\left\{y_0\right\}\right)\right)\cap \left(-C\setminus \left\{0_Y\right\}\right).$$

By the strong monotonicity in Lemma 2, we have

$$\alpha \Vert y^{*}\Vert +l\left(y^{*}\right)<0$$

which contradicts (20) and this proves the theorem. □

Theorem 3. 

Let (X,‖.‖_X) and (Y,‖.‖_Y) be real normed spaces, let C ⊂ Y be a closed pointed convex cone with int(C) ≠ ∅, let F:X → 2^Y be a set-valued map. Let a pair $\left(\overline{x},\overline{y}\right)\in \mathit{graph}\left(F\right)$ be given. Assume that for some x ∈ X the weak generalized radial epiderivative $D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$ exists. Then

$$\bigcup _{(l,\alpha )\in U^{⧣}}{\mathit{SG}}_{(l,\alpha )}\left(x\right)\subseteq D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$$

Proof. 

Let $y_0\in \bigcup _{\left(l,\alpha \right)\in U^{⧣}}S{G}_{\left(l,\alpha \right)}\left(x\right)$. Then there exists (l,α) ∈ U^⧣ such that for every y ∈ G(x)

$$\alpha ‖y_0‖+l\left(y_0\right)\le \alpha ‖y‖+l\left(y\right)$$

(21)

By assumption the weak generalized radial epiderivative $D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$ exists. Since $D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)=w-\mathit{min}\left(G\left(x\right),C\right)$, w − min(G(x),C) ≠ ∅ and therefore G(x) ≠ ∅. Let $y_0\notin D_{\mathit{wr}}F\left(\overline{x},\overline{y}\right)\left(x\right)$. Then there exists some

$$y^{*}\in G\left(x\right)\cap \left(y_0-\mathit{int}\left(C\right)\right).$$

By Lemma 2, we have

$$\alpha ‖y^{*}‖+l\left(y^{*}\right)<\alpha ‖y_0‖+l\left(y_0\right),$$

which contradicts (21) and the theorem is proved. □

4. Conclusions

In this paper, we have extended the generalized radial epiderivative framework for nonconvex set-valued maps by introducing two new notions: the weak generalized radial epiderivative and the proper generalized radial epiderivative. These concepts are obtained by using weak minimality and Benson proper minimality, respectively, in the definition of the generalized radial epiderivative.

We have established existence results for the generalized radial epiderivatives and clarified their relationships with the radial epiderivative. In particular, sufficient conditions based on the properties of the ordering cone and the domination property have been given. We have also obtained scalarization-based characterizations of the proper and weak generalized radial epiderivatives by using strongly monotonically increasing functionals. The examples presented in the paper illustrate that the proposed notions may differ from the generalized radial epiderivative and therefore provide genuinely distinct extensions of the existing framework.

These results provide a basis for further investigations of generalized differentiation and optimality conditions in nonconvex set-valued optimization. Future work may focus on applications of the proposed epiderivatives to vector and set-valued optimization problems and on the development of corresponding optimality conditions.